Module 09
library(tidyverse)
library(broom) # For tidying model output
library(leaps) # For best subset selection
library(readxl) # For reading Excel files
library(patchwork) # For combining plots
library(latex2exp) # For LaTeX expressions
library(rlang) # For working with formulas
library(glmnet) # For ridge regression and the lasso Applied Statistical Techniques
For a linear model with \(p\) predictors
\[ Y = \beta_0 + \beta_1X_1 + \beta_2X_2 + \ldots + \beta_pX_p + \epsilon \]
there are \(2^p\) possible models.
For example with \(p=3\) predictors, there are \(2^3 = 8\) possible models. They can be expressed as:
\[ \begin{align*} Y &= \beta_0 + \beta_1X_1 + \beta_2X_2 + \beta_3X_3 + \epsilon \\ Y &= \beta_0 + \beta_1X_1 + \beta_2X_2 + \epsilon \\ Y &= \beta_0 + \beta_1X_1 + \beta_3X_3 + \epsilon \\ Y &= \beta_0 + \beta_2X_2 + \beta_3X_3 + \epsilon \\ Y &= \beta_0 + \beta_1X_1 + \epsilon \\ Y &= \beta_0 + \beta_2X_2 + \epsilon \\ Y &= \beta_0 + \beta_3X_3 + \epsilon \\ Y &= \beta_0 + \epsilon \\ \end{align*} \]
Prediction Accuracy
Constraints on the number of predictors can lead to better prediction accuracy.
Model Interpretability
A model with fewer predictors is easier to interpret.
Note:
Step 2 is preformed on a training set.
Step 3 is performed on a validation set.
\[ \binom{p}{k} = (\textrm{p choose k})= \frac{p!}{k!(p-k)!} \]
Variable Information:
Given is the variable name, variable type, the measurement unit and a brief description. The concrete compressive strength is the regression problem. The order of this listing corresponds to the order of numerals along the rows of the database.
Name – Data Type – Measurement – Description
Subset selection object
Call: regsubsets.formula(strength ~ ., data = concrete, nvmax = 8)
8 Variables (and intercept)
Forced in Forced out
cement FALSE FALSE
slag FALSE FALSE
ash FALSE FALSE
water FALSE FALSE
superplasticizer FALSE FALSE
coarse_aggregate FALSE FALSE
fine_aggregate FALSE FALSE
age FALSE FALSE
1 subsets of each size up to 8
Selection Algorithm: exhaustive
cement slag ash water superplasticizer coarse_aggregate fine_aggregate
1 ( 1 ) "*" " " " " " " " " " " " "
2 ( 1 ) "*" " " " " " " "*" " " " "
3 ( 1 ) "*" " " " " " " "*" " " " "
4 ( 1 ) "*" "*" " " "*" " " " " " "
5 ( 1 ) "*" "*" "*" "*" " " " " " "
6 ( 1 ) "*" "*" "*" "*" "*" " " " "
7 ( 1 ) "*" "*" "*" "*" "*" "*" " "
8 ( 1 ) "*" "*" "*" "*" "*" "*" "*"
age
1 ( 1 ) " "
2 ( 1 ) " "
3 ( 1 ) "*"
4 ( 1 ) "*"
5 ( 1 ) "*"
6 ( 1 ) "*"
7 ( 1 ) "*"
8 ( 1 ) "*"Subset selection object
Call: regsubsets.formula(strength ~ ., data = concrete, nvmax = 8,
nbest = 256, really.big = TRUE)
8 Variables (and intercept)
Forced in Forced out
cement FALSE FALSE
slag FALSE FALSE
ash FALSE FALSE
water FALSE FALSE
superplasticizer FALSE FALSE
coarse_aggregate FALSE FALSE
fine_aggregate FALSE FALSE
age FALSE FALSE
256 subsets of each size up to 8
Selection Algorithm: exhaustive
cement slag ash water superplasticizer coarse_aggregate
1 ( 1 ) "*" " " " " " " " " " "
1 ( 2 ) " " " " " " " " "*" " "
1 ( 3 ) " " " " " " " " " " " "
1 ( 4 ) " " " " " " "*" " " " "
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1 ( 8 ) " " " " "*" " " " " " "
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5 ( 11 ) "*" "*" "*" " " " " "*"
5 ( 12 ) " " " " "*" "*" " " "*"
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5 ( 14 ) "*" " " "*" "*" "*" " "
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5 ( 16 ) "*" " " " " " " "*" "*"
5 ( 17 ) "*" " " "*" "*" " " " "
5 ( 18 ) " " "*" " " "*" " " "*"
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5 ( 25 ) "*" "*" "*" "*" " " "*"
5 ( 26 ) "*" "*" "*" " " "*" "*"
5 ( 27 ) "*" "*" " " "*" "*" " "
5 ( 28 ) "*" "*" "*" " " "*" " "
5 ( 29 ) "*" "*" " " "*" " " "*"
5 ( 30 ) "*" "*" "*" " " " " "*"
5 ( 31 ) "*" "*" " " "*" "*" "*"
5 ( 32 ) "*" "*" " " " " " " "*"
5 ( 33 ) "*" "*" " " " " "*" "*"
5 ( 34 ) "*" " " "*" "*" " " "*"
5 ( 35 ) " " "*" " " "*" "*" " "
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5 ( 37 ) " " " " "*" " " "*" "*"
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5 ( 39 ) " " "*" "*" " " "*" " "
5 ( 40 ) "*" " " "*" "*" "*" " "
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5 ( 43 ) " " "*" "*" "*" " " "*"
5 ( 44 ) "*" " " "*" " " "*" "*"
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5 ( 46 ) "*" " " "*" " " " " "*"
5 ( 47 ) " " "*" " " "*" "*" "*"
5 ( 48 ) "*" " " "*" "*" "*" "*"
5 ( 49 ) " " "*" "*" "*" " " "*"
5 ( 50 ) " " "*" "*" " " "*" "*"
5 ( 51 ) " " "*" " " " " "*" "*"
5 ( 52 ) " " "*" " " "*" "*" "*"
5 ( 53 ) " " "*" "*" "*" "*" " "
5 ( 54 ) " " "*" "*" " " "*" "*"
5 ( 55 ) " " "*" "*" "*" "*" "*"
5 ( 56 ) " " "*" "*" " " " " "*"
6 ( 1 ) "*" "*" "*" "*" "*" " "
6 ( 2 ) "*" "*" "*" "*" " " " "
6 ( 3 ) "*" "*" "*" "*" " " "*"
6 ( 4 ) "*" "*" "*" " " " " "*"
6 ( 5 ) "*" "*" " " "*" " " "*"
6 ( 6 ) "*" "*" " " "*" "*" " "
6 ( 7 ) "*" "*" "*" " " "*" "*"
6 ( 8 ) "*" "*" " " "*" "*" "*"
6 ( 9 ) "*" "*" "*" " " "*" " "
6 ( 10 ) "*" " " "*" "*" " " "*"
6 ( 11 ) "*" " " " " "*" "*" "*"
6 ( 12 ) "*" "*" " " " " "*" "*"
6 ( 13 ) " " "*" "*" "*" " " "*"
6 ( 14 ) "*" " " "*" "*" "*" " "
6 ( 15 ) " " " " "*" "*" "*" "*"
6 ( 16 ) "*" " " "*" "*" "*" "*"
6 ( 17 ) "*" " " "*" " " "*" "*"
6 ( 18 ) " " "*" " " "*" "*" "*"
6 ( 19 ) " " "*" "*" "*" "*" " "
6 ( 20 ) "*" "*" "*" "*" "*" "*"
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6 ( 22 ) "*" "*" "*" " " "*" "*"
6 ( 23 ) "*" "*" "*" "*" " " "*"
6 ( 24 ) "*" "*" " " "*" "*" "*"
6 ( 25 ) "*" " " "*" "*" "*" "*"
6 ( 26 ) " " "*" "*" " " "*" "*"
6 ( 27 ) " " "*" "*" "*" "*" "*"
6 ( 28 ) " " "*" "*" "*" "*" "*"
7 ( 1 ) "*" "*" "*" "*" "*" "*"
7 ( 2 ) "*" "*" "*" "*" "*" " "
7 ( 3 ) "*" "*" "*" "*" " " "*"
7 ( 4 ) "*" "*" "*" " " "*" "*"
7 ( 5 ) "*" "*" " " "*" "*" "*"
7 ( 6 ) "*" " " "*" "*" "*" "*"
7 ( 7 ) " " "*" "*" "*" "*" "*"
7 ( 8 ) "*" "*" "*" "*" "*" "*"
8 ( 1 ) "*" "*" "*" "*" "*" "*"
fine_aggregate age
1 ( 1 ) " " " "
1 ( 2 ) " " " "
1 ( 3 ) " " "*"
1 ( 4 ) " " " "
1 ( 5 ) "*" " "
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6 ( 28 ) "*" " "
7 ( 1 ) " " "*"
7 ( 2 ) "*" "*"
7 ( 3 ) "*" "*"
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7 ( 7 ) "*" "*"
7 ( 8 ) "*" " "
8 ( 1 ) "*" "*"
Call:
lm(formula = strength ~ cement, data = concrete)
Residuals:
Min 1Q Median 3Q Max
-40.594 -10.952 -0.572 9.992 43.241
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 13.442795 1.296925 10.37 <2e-16 ***
cement 0.079580 0.004324 18.41 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 14.5 on 1028 degrees of freedom
Multiple R-squared: 0.2478, Adjusted R-squared: 0.2471
F-statistic: 338.7 on 1 and 1028 DF, p-value: < 2.2e-16
Call:
lm(formula = strength ~ age, data = concrete)
Residuals:
Min 1Q Median 3Q Max
-38.509 -11.290 -1.516 9.426 47.469
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 31.846436 0.606949 52.47 <2e-16 ***
age 0.086974 0.007789 11.17 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 15.78 on 1028 degrees of freedom
Multiple R-squared: 0.1082, Adjusted R-squared: 0.1073
F-statistic: 124.7 on 1 and 1028 DF, p-value: < 2.2e-16regsubsets() function?(You will need to review the documentation for details.)
which: A logical matrix indicating which elements are in each model
rsq: The r-squared for each model
rss: Residual sum of squares for each model
adjr2: Adjusted r-squared
cp: Mallows’ Cp
bic: Schwartz’s information criterion, BIC
outmat: A version of the which component that is formatted for printing
obj: A copy of the regsubsets object
concrete_subset_adjr2 <- as_tibble(summary(regfit_full)$which, rownames = "model") |>
select(model) |>
bind_cols(as_tibble_col(summary(regfit_full)$adjr2, column_name = "adjr2")) |>
mutate(model = as_factor(model))
concete_model_max_adjr2 <- concrete_subset_adjr2 |>
group_by(model) |>
summarise(adjr2 = max(adjr2)) |>
ungroup()
concrete_subset_adjr2 |>
ggplot() +
geom_point(aes(x = model, y = adjr2), alpha = 1/5) +
geom_line(aes(x = model, y = adjr2, group = 1),
data = concete_model_max_adjr2,
color = "red") +
geom_point(aes(x = model, y = adjr2),
data = concete_model_max_adjr2,
color = "red") +
labs(
title = "Best Subset Selection for Concrete Data",
subtitle = "Adjusted *R<sup>2</sup>* for each model",
x = "Number of Predictors",
y = NULL
) +
theme_light() +
theme(
plot.title = ggtext::element_markdown(size = 16),
plot.title.position = "plot",
plot.subtitle = ggtext::element_markdown(size = 12)
)
Subset selection object
Call: regsubsets.formula(strength ~ ., data = concrete, nvmax = 8,
really.big = FALSE)
8 Variables (and intercept)
Forced in Forced out
cement FALSE FALSE
slag FALSE FALSE
ash FALSE FALSE
water FALSE FALSE
superplasticizer FALSE FALSE
coarse_aggregate FALSE FALSE
fine_aggregate FALSE FALSE
age FALSE FALSE
1 subsets of each size up to 8
Selection Algorithm: exhaustive
cement slag ash water superplasticizer coarse_aggregate fine_aggregate
1 ( 1 ) "*" " " " " " " " " " " " "
2 ( 1 ) "*" " " " " " " "*" " " " "
3 ( 1 ) "*" " " " " " " "*" " " " "
4 ( 1 ) "*" "*" " " "*" " " " " " "
5 ( 1 ) "*" "*" "*" "*" " " " " " "
6 ( 1 ) "*" "*" "*" "*" "*" " " " "
7 ( 1 ) "*" "*" "*" "*" "*" "*" " "
8 ( 1 ) "*" "*" "*" "*" "*" "*" "*"
age
1 ( 1 ) " "
2 ( 1 ) " "
3 ( 1 ) "*"
4 ( 1 ) "*"
5 ( 1 ) "*"
6 ( 1 ) "*"
7 ( 1 ) "*"
8 ( 1 ) "*"as_tibble(summary(regfit_full_8)$which, rownames = "model") |>
mutate(
adjr2 = summary(regfit_full_8)$adjr2,
model = as_factor(model),
) |>
select(-`(Intercept)`) |>
pivot_longer(cols = 2:9, , names_to = "variable", values_to = "included") |>
mutate(variable = str_replace(variable, "_", " ")) |>
mutate(variable = reorder(as_factor(variable), adjr2)) |>
ggplot() +
geom_tile(aes(x = model, y = variable, fill = included), color = "white") +
labs(
title = "Best Subset Selection for Concrete Data",
subtitle = "Variables available for each model ordered by adjusted *R<sup>2</sup>*",
x = "Number of Predictors",
y = NULL,
fill = "Varible included \nin the model?"
) +
scale_fill_manual(values = c("red", "darkgrey")) +
scale_x_discrete() +
theme_minimal() +
theme(
plot.title = ggtext::element_markdown(size = 16),
plot.title.position = "plot",
plot.subtitle = ggtext::element_markdown(size = 12)
) 
Subset selection is a useful tool for selecting the best model for a given data set; however, it can be computationally expensive.
If the number of predictors is large, the chance of overfitting the model also increases.
We can use Stepwise methods to evaluate models with a large number of predictors.
Forward selection starts with the null model and adds predictors one at a time. The number of models fit is
\[ 1 + p(p+1)/2 \]
For our example of \(p = 8\) predictors, this would be \(1 + 8(8+1)/2 = 37\) models instead of the \(2^8 = 256\) models for best subset selection.
Backward selection starts with the full model and removes predictors one at a time. The number of models fit is the same as for forward selection.
Subset selection object
Call: regsubsets.formula(strength ~ ., data = concrete, nvmax = 8,
method = "forward")
8 Variables (and intercept)
Forced in Forced out
cement FALSE FALSE
slag FALSE FALSE
ash FALSE FALSE
water FALSE FALSE
superplasticizer FALSE FALSE
coarse_aggregate FALSE FALSE
fine_aggregate FALSE FALSE
age FALSE FALSE
1 subsets of each size up to 8
Selection Algorithm: forward
cement slag ash water superplasticizer coarse_aggregate fine_aggregate
1 ( 1 ) "*" " " " " " " " " " " " "
2 ( 1 ) "*" " " " " " " "*" " " " "
3 ( 1 ) "*" " " " " " " "*" " " " "
4 ( 1 ) "*" "*" " " " " "*" " " " "
5 ( 1 ) "*" "*" " " "*" "*" " " " "
6 ( 1 ) "*" "*" "*" "*" "*" " " " "
7 ( 1 ) "*" "*" "*" "*" "*" "*" " "
8 ( 1 ) "*" "*" "*" "*" "*" "*" "*"
age
1 ( 1 ) " "
2 ( 1 ) " "
3 ( 1 ) "*"
4 ( 1 ) "*"
5 ( 1 ) "*"
6 ( 1 ) "*"
7 ( 1 ) "*"
8 ( 1 ) "*"Subset selection object
Call: regsubsets.formula(strength ~ ., data = concrete, nvmax = 8,
method = "backward")
8 Variables (and intercept)
Forced in Forced out
cement FALSE FALSE
slag FALSE FALSE
ash FALSE FALSE
water FALSE FALSE
superplasticizer FALSE FALSE
coarse_aggregate FALSE FALSE
fine_aggregate FALSE FALSE
age FALSE FALSE
1 subsets of each size up to 8
Selection Algorithm: backward
cement slag ash water superplasticizer coarse_aggregate fine_aggregate
1 ( 1 ) "*" " " " " " " " " " " " "
2 ( 1 ) "*" " " " " " " " " " " " "
3 ( 1 ) "*" " " " " "*" " " " " " "
4 ( 1 ) "*" "*" " " "*" " " " " " "
5 ( 1 ) "*" "*" "*" "*" " " " " " "
6 ( 1 ) "*" "*" "*" "*" "*" " " " "
7 ( 1 ) "*" "*" "*" "*" "*" "*" " "
8 ( 1 ) "*" "*" "*" "*" "*" "*" "*"
age
1 ( 1 ) " "
2 ( 1 ) "*"
3 ( 1 ) "*"
4 ( 1 ) "*"
5 ( 1 ) "*"
6 ( 1 ) "*"
7 ( 1 ) "*"
8 ( 1 ) "*"Important
The model containing all of the predictors will always have the smallest \(RSS\) and the largest \(R^2\)
Recall, we want to choose a model based on its ability to predict new data.
In the above examples, we didn’t create a test set to evaluate the models, we simply used the entire dataset.
Unfortunately, the regsubsets() function does not have a built-in method for evaluating models on a test set.
\[ RSS = \sum_{i=1}^{n} (y_i - \hat{y}_i)^2 \]
\[ R^2 = 1 - \frac{RSS}{TSS} \]
\(TSS = \sum_{i=1}^{n} (y_i - \bar{y})^2\) is the Total Sum of Squares.
\[ MSE = \frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2 = \frac{RSS}{n} \]
\[ \textrm{Adjusted }R^2 = 1 - \frac{RSS/(n-d-1)}{TSS/(n-1)} \]
\[ C_p = \frac{1}{n} (RSS + 2d\hat{\sigma}^2) \]
\[ AIC = \frac{1}{n} (RSS + 2d\hat{\sigma}^2) \]
\[ BIC = \frac{1}{n} (RSS + \log(n)d\hat{\sigma}^2) \]
where \(d\) is the number of predictors in the model and \(\hat{\sigma}^2\) is an estimate of the variance of the error term.
Walk through the following code to examing how we can work with combinations of predictors.
# Initialize an empty list to store model information
model_list <- list()
total_combinations <- 0
for (i in 1:8) {
concrete_predictors <- combn(concrete_column_names[1:8], i)
num_combinations <- ncol(concrete_predictors)
cat(sprintf("\nNumber of predictors: %d\n", i))
cat(sprintf("Number of combinations: %d\n", num_combinations))
total_combinations <- total_combinations + num_combinations
for (j in 1:num_combinations) {
formula <- reformulate(concrete_predictors[,j], response = "strength")
model_combination <- lm(formula, data = concrete)
# Add model information to the list
model_list[[length(model_list) + 1]] <- list(
num_predictors = i,
combination = paste(concrete_predictors[,j], collapse = ", "),
formula = paste(deparse(formula), collapse = " "), # Ensure single string
model = model_combination
)
}
}
Number of predictors: 1
Number of combinations: 8
Number of predictors: 2
Number of combinations: 28
Number of predictors: 3
Number of combinations: 56
Number of predictors: 4
Number of combinations: 70
Number of predictors: 5
Number of combinations: 56
Number of predictors: 6
Number of combinations: 28
Number of predictors: 7
Number of combinations: 8
Number of predictors: 8
Number of combinations: 1# Convert the list to a tibble
model_results <- tibble::tibble(
num_predictors = map_int(model_list, "num_predictors"),
combination = map_chr(model_list, "combination"),
formula = map_chr(model_list, "formula"),
model = map(model_list, "model")
)
cat(sprintf("\nTotal number of combinations: %d\n", total_combinations))
Total number of combinations: 255Number of rows in model_results: 255For many cases, you’ll be able to use package functions to perform model selection.
# Created with the help of Calude.ai
compare_models <- function(data, response_var, predictor_vars, proportion = 0.8, seed = NULL) {
# Set seed if provided
if (!is.null(seed)) set.seed(seed)
# Capture the predictor variables
pred_vars <- enquo(predictor_vars)
# Select the specified columns
selected_data <- data |>
select(!!response_var, !!pred_vars)
# Get column names
all_column_names <- colnames(selected_data)
response_name <- as.character(response_var)
predictor_names <- setdiff(all_column_names, response_name)
# Create train/test split
n <- nrow(selected_data)
train_indices <- sample(1:n, size = round(proportion * n))
train_data <- selected_data[train_indices, ]
test_data <- selected_data[-train_indices, ]
# Initialize an empty list to store model information
model_list <- list()
total_combinations <- 0
for (i in 1:length(predictor_names)) {
predictor_combinations <- combn(predictor_names, i)
num_combinations <- ncol(predictor_combinations)
total_combinations <- total_combinations + num_combinations
for (j in 1:num_combinations) {
formula <- reformulate(predictor_combinations[,j], response = response_name)
model_combination <- lm(formula, data = train_data)
# Make predictions on test data
predictions <- predict(model_combination, newdata = test_data)
# Calculate RMSE and R-squared on test data
rmse <- sqrt(mean((test_data[[response_name]] - predictions)^2))
rsq <- cor(test_data[[response_name]], predictions)^2
# Add model information to the list
model_list[[length(model_list) + 1]] <- list(
num_predictors = i,
combination = paste(predictor_combinations[,j], collapse = ", "),
formula = paste(deparse(formula), collapse = " "),
model = model_combination,
test_rmse = rmse,
test_rsq = rsq
)
}
}
# Convert the list to a tibble
model_results <- tibble::tibble(
num_predictors = map_int(model_list, "num_predictors"),
combination = map_chr(model_list, "combination"),
formula = map_chr(model_list, "formula"),
model = map(model_list, "model"),
test_rmse = map_dbl(model_list, "test_rmse"),
test_rsq = map_dbl(model_list, "test_rsq")
)
cat(sprintf("\nTotal number of models: %d\n", nrow(model_results)))
return(model_results)
}
# Example usage:
result <- compare_models(concrete, "strength", concrete_column_names, proportion = 0.8, seed = 42)
Total number of models: 255max_results <- result |>
group_by(num_predictors) |>
filter(test_rsq == max(test_rsq)) |>
ungroup()
result |>
ggplot() +
geom_point(aes(x = num_predictors, y = test_rsq), alpha = 1/3) +
geom_line(aes(x = num_predictors, y = test_rsq), data = max_results, color = "red") +
geom_point(aes(x = num_predictors, y = test_rsq), data = max_results, color = "red") +
labs(
title = "Model Selection Using Subset Selection on Test Data",
subtitle = "Custom function to compare models based on test data",
x = "Number of Predictors",
y = "R<sup>2</sup>"
) +
scale_x_continuous(breaks = 1:8) +
theme_light() +
theme(
plot.title = ggtext::element_markdown(size = 16),
plot.title.position = "plot",
plot.subtitle = ggtext::element_markdown(size = 12),
axis.title = ggtext::element_markdown(size = 10)
)Recall that linear regression minimizes RSS to find the best-fitting line.
\[ RSS = \sum_{i=1}^{n} (y_i - \hat{y}_i)^2 = \sum_{i=1}^{n} \left (y_i - \beta_0 - \sum_{j=1}^{p} \beta_j x_{ij} \right )^2 \]
Ridge regression adds a penalty term to the RSS to shrink the coefficients towards zero.
\[ \sum_{i=1}^{n} \left (y_i - \beta_0 - \sum_{j=1}^{p} \beta_j x_{ij} \right )^2 + \lambda \sum_{j=1}^{p} \beta_j^2 \]
\(\lambda\) controls the amount of shrinkage.
\(\lambda \ge 0\) is a tuning parameter that controls the amount of shrinkage and is determined using cross-validation.
Note
We will use the glmnet package to perform ridge regression.
The penalty term is often referred to as l2 regularization and has a shorthand notation of
\[ || \beta ||_2^2 = \sum_{j=1}^{p} \beta_j^2 \]
The ridge regression coefficient estimates can change substantially when multiplying a given predictor by a constant.
\(X_j \hat{\beta}_{j,\lambda}^R\) will depend not only on the value of \(\lambda\), but also on the scaling of the \(j\)th predictor. . . the value of \(X_j \hat{\beta}_{j,\lambda}^R\) may even depend on the scaling of the other predictors! Therefore, it is best to apply ridge regression after standardizing the predictors, using the formula
\[ \tilde{x}_{ij} = \frac{x_{ij}}{\sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_{ij} - \bar{x}_j)^2}} \]
Alternatively, we can use the scale() function in R to center and scale the predictors.
This is often referred to as z-score scaling.
\[ \tilde{x}_{ij} = \frac{x_{ij} - \bar{x}_j}{\sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_{ij} - \bar{x}_j)^2}} \]
concrete_ridge <- glmnet(x = scale(as.matrix(concrete[, 1:8])), y = concrete$strength, alpha = 0, standardize = FALSE)
# names(concrete_ridge)
# concrete_ridge$lambda
as_tibble(as.matrix(concrete_ridge$beta), rownames = "predictor") |>
pivot_longer(cols = 2:101, names_to = "lambda", values_to = "beta_values") |>
mutate(lambda_values = rep(concrete_ridge$lambda, times = 8)) |>
ggplot() +
geom_line(aes(x = lambda_values, y = beta_values, color = predictor)) +
labs(
title = "Ridge Regression Coefficients",
subtitle = "Coefficient values for each predictor over different lambda values",
x = TeX(r'($lambda$)'),
y = "Coefficient Value"
) +
scale_x_log10() +
theme_light()
lambda_values <- 10^seq(3, -1, length = 100)
concrete_ridge_cv <- cv.glmnet(x = scale(as.matrix(concrete[, 1:8])), y = concrete$strength, alpha = 0, standardize = FALSE, nfolds = 10, lambda = lambda_values)
ggplot() +
geom_point(aes(x = concrete_ridge_cv$lambda, y = concrete_ridge_cv$cvm)) +
labs(
title = "Ridge Regression Cross-Validation",
subtitle = "Cross-validated mean squared error for different lambda values",
x = TeX(r'($\lambda$)'),
y = "Mean Squared Error"
) +
scale_x_log10(label = scales::label_comma(accuracy = 1)) +
theme_light()
\[ \sum_{i=1}^{n} \left (y_i - \beta_0 - \sum_{j=1}^{p} \beta_j x_{ij} \right )^2 + \lambda \sum_{j=1}^{p} | \beta_j | \]
Similar to Ridge regression, \(\lambda \ge 0\) is a tuning parameter that controls the amount of shrinkage and is determined using cross-validation.
Unlike Ridge regression, the lasso can set coefficients to zero, effectively performing variable selection.
The data should be standardized before using the lasso.
concrete_lasso <- glmnet(x = scale(as.matrix(concrete[, 1:8])), y = concrete$strength, alpha = 1, standardize = FALSE)
as_tibble(as.matrix(concrete_lasso$beta), rownames = "predictor") |>
pivot_longer(cols = 2:78, names_to = "lambda", values_to = "beta_values") |>
mutate(lambda_values = rep(concrete_lasso$lambda, times = 8)) |>
ggplot() +
geom_line(aes(x = lambda_values, y = beta_values, color = predictor)) +
labs(
title = "Lasso Coefficients",
subtitle = "Coefficient values for each predictor over different lambda values",
x = TeX(r'($\lambda$)'),
y = "Coefficient Value"
) +
scale_x_log10() +
theme_light()
lambda_values <- 10^seq(2, -1, length = 100)
concrete_lasso_cv <- cv.glmnet(x = scale(as.matrix(concrete[, 1:8])), y = concrete$strength, alpha = 1, standardize = FALSE, nfolds = 10, lambda = lambda_values)
ggplot() +
geom_point(aes(x = concrete_lasso_cv$lambda, y = concrete_lasso_cv$cvm)) +
labs(
title = "Lasso Cross-Validation",
subtitle = "Cross-validated mean squared error for different lambda values",
x = TeX(r'($\lambda$)'),
y = "Mean Squared Error"
) +
scale_x_log10() +
theme_light()
\[ \sum_{i=1}^{n} \left (y_i - \beta_0 - \sum_{j=1}^{p} \beta_j x_{ij} \right )^2 + \lambda \left ( \alpha \sum_{j=1}^{p} | \beta_j | + (1 - \alpha) \sum_{j=1}^{p} \beta_j^2 \right ) \]
where \(\alpha\) controls the balance between Ridge and Lasso regression.
\(\alpha = 1\) is Lasso regression and is the default in glmnet. \(\alpha = 0\) is Ridge regression.
We can extend linear regression to include polynomial terms where the coefficients are still linear, but the predictors are polynomial terms of the original predictors.
\[ y_i = \beta_0 + \beta_1 x_i + \beta_2 x_i^2 + \ldots + \beta_d x_i^d + \epsilon_i \]
In most cases, the value of \(d\) is small, often 2 or 3 as higher values can lead to over fitting.
Call:
lm(formula = Response ~ `Load Level`, data = loadcell)
Residuals:
Min 1Q Median 3Q Max
-3.181e-04 -2.075e-04 -2.862e-05 1.983e-04 4.208e-04
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -7.156e-04 9.025e-05 -7.929 5.97e-09 ***
`Load Level` 1.003e-01 6.725e-06 14909.755 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.000239 on 31 degrees of freedom
Multiple R-squared: 1, Adjusted R-squared: 1
F-statistic: 2.223e+08 on 1 and 31 DF, p-value: < 2.2e-16Note: R-squared is 1!
Call:
lm(formula = Response ~ poly(`Load Level`, 2), data = loadcell)
Residuals:
Min 1Q Median 3Q Max
-9.966e-05 -1.466e-05 5.944e-06 2.515e-05 5.595e-05
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.193e+00 6.552e-06 182130.14 <2e-16 ***
poly(`Load Level`, 2)1 3.563e+00 3.764e-05 94658.88 <2e-16 ***
poly(`Load Level`, 2)2 1.314e-03 3.764e-05 34.92 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 3.764e-05 on 30 degrees of freedom
Multiple R-squared: 1, Adjusted R-squared: 1
F-statistic: 4.48e+09 on 2 and 30 DF, p-value: < 2.2e-16set.seed(42) # added so jitter will not change between renders
loadcell_resid_quad <- augment(loadcell_lm_quad)
loadcell_resid_quad |>
ggplot() +
geom_point(aes(x = loadcell$`Load Level`, y = .resid),
position = position_jitter(width = 0.15),
shape = 1) +
labs(
title = "Loadcell Residuals from Quadratic Regression",
subtitle = "Residuals are randomly distributed",
x = "Load Level",
y = "Residuals"
) +
theme_light()
Call:
lm(formula = Response ~ `Load Level` + I(`Load Level`^2), data = loadcell)
Residuals:
Min 1Q Median 3Q Max
-9.966e-05 -1.466e-05 5.944e-06 2.515e-05 5.595e-05
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.840e-05 2.451e-05 -0.751 0.459
`Load Level` 1.001e-01 4.839e-06 20687.891 <2e-16 ***
I(`Load Level`^2) 7.032e-06 2.014e-07 34.922 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 3.764e-05 on 30 degrees of freedom
Multiple R-squared: 1, Adjusted R-squared: 1
F-statistic: 4.48e+09 on 2 and 30 DF, p-value: < 2.2e-16